# arrow_back Prove that if $X$ is a random variable with mean $E X$ and $c \in$ is $a$ real number, then $\operatorname{Var}(X)=E\left(X^{2}\right)-E(X)$

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Prove that if $X$ is a random variable with mean $E X$ and $c \in$ is $a$ real number, then $\operatorname{Var}(X)=E\left(X^{2}\right)-E(X)$

Expanding out the square in the definition of variance gives
\begin{aligned} \operatorname{Var}(X) &=E\left[(X-E X)^{2}\right] \\ &=E\left[X^{2}-2 X E X+(E X)^{2}\right] \\ &=E\left[X^{2}\right]-E(2 X E X)+E\left((E X)^{2}\right) \\ &=E\left[X^{2}\right]-2 E X E X+(E X)^{2} \\ &=E\left[X^{2}\right]-(E X)^{2} \end{aligned}
where the third equality comes from linearity of $E$ (Exercise $2.3$
(a)) and the fourth equality comes from Exercise $2.3(\mathrm{~b})$ and the fact that since $E X$ and $(E X)^{2}$ are constants, their expectations are just $E X$ and $(E X)^{2}$ respectively.
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